Geometry group theory is the study of geometric objects and their groups of symmetries using mathematical tools such as algebra, topology, and analysis, with a special emphasis on "large scale" or "quasi-isometric" tools which are used in studies of a geometric object as if viewing it through more and more powerful telescopes. In this project the PI will apply the methods of geometric group theory to the study of hidden symmetries of finite networks known to topologists as "graphs". These groups of hidden symmetries are known as "outer automorphism groups of free groups". One outcome of this project will be to solve important problems in the large scale geometry of outer automorphism groups of free groups and through these solutions to produce new tools of large scale geometry for broad use by the geometric group theory community. Another outcome will be to build tools for a new study of the dynamical properties of outer automorphism groups of free groups.
Outer automorphism groups of finite rank free groups, denoted Out(F_n), were first examined in the 1930's by Nielsen and Whitehead. Interest in Out(F_n) has expanded rapidly in recent decades with the development of geometric tools such as: the Culler-Vogtmann outer space; and the relative train track maps of Bestvina-Feighn-Handel. More recently Out(F_n) is being studied using quasi-isometric geometry, an effort in which the principal investigator has played a leading role. In this project the PI will study actions of finitely generated subgroups of Out(F_n) on hyperbolic metric spaces, attempting to build for Out(F_n) an analogue of the hierarchy theory for mapping class groups of surfaces that was built in the 1990's by Masur and Minsky. As applications, and as guides to building a useful hierarchy theory, the PI plans to study and resolve various problems in the large scale geometry of Out(F_n) including: computation of the 2nd bounded cohomology of finitely generated subgroups of Out(F_n); and early investigations into quasi-isometric rigidity for Out(F_n). The PI also plans to carry out early investigations of certain Out(F_n)-invariant dynamical systems that are analogous to the Teichmuller geodesic flow of a surface and its invariant sets.
